Curriculum Queries

 

I do not boast lots of boots on the ground, inventing tests and tallying data. I might welcome such an army, using Python and Jupyter Notebooks perhaps. I share about pyplot and plotly.

I'm speaking with reference to such queries as where best to introduce figurate numbers, such as triangular and square, if we do, and when to make those polyhedral (icosahedral, cuboctahedral, tetrahedral...). I'm eyeing alternative vocabs. Sometimes it's the animation that matters, more than the script rendering language, a namespace.

What am I talking about pray tell? Like the numbers that go 1, 2, 3, 4, 5... then keep accumulating: 1, 3, 6, 10, 15... it's like scooping up the snow: all the layers of balls (if we think in balls) make us a growing triangle.  The triangular numbers: 1, 3, 6, 10, 15... Then we have the squares: 1, 4, 9, 16, 25... You know the ones, right?

We've got that going on, and then we layer-pack around a nuclear ball, per a cuboctahedral shape (cube of sawed-off corners), thereby getting our famous "successive layers" sequence: 1, 12, 42, 92, 162... I say "famous" because it literally is in the center of our neighborhood (along with the cumulative Crystal Ball sequence). "We have statues dedicated to it" one could claim, especially if coining an idiom. So where does all that fit in?

"Nowhere!" comes the resounding (highly opinionated) voice from some quarters. Curriculum developers don't just accept these queries lying down, without some furious debate. 

I point to The Book of Numbers by Conway and Guy and maybe that mollifies some (ah, mainstream). I reassure teachers that we're connecting right brained shapes with left brain numbers even earlier than by means of coordinate framework addressing ala XYZ. 

We haven't even come to XYZ coordinates yet and we're already getting rhombic dodecahedra in their space-filling role. "Are we to expect IVM coordinates then?" sounds sardonic but yes, we have those for you, why not be curious?

Then finally comes this question whether we want to not only disclose the ratios here, but harp on them, taking sphere packing as a home base worth fighting for. We shift our balance from the XYZ to IVM framework for a change. What change? That remains to be beholden, for the most part, but as someone shifting my balance, I'm suggesting it's not that hard, nor some kind of one way street.  You're free to switch back and forth. I do, routinely.

What am I talking about again, am I making any kind of sense? That depends to some degree on what archeological layer you're reading from. 

Those in the immediate radius of Buckminster Fuller (not me, I came later), in his role as academic, would recognize "isotropic vector matrix" in many cases, especially in the late 1970s after Synergetics was published. I became aware of Fuller's philosophical language in the early 1980s, as I switched gears from being a philosophy major at Princeton (Rorty, Kaufmann... other stars), to being a high school teacher in a private school in Jersey City (not that far from Princeton actually, by dinky, Amtrak and PATH).

The idea is one of scaffolding or tiling in space. The all-cubes way of filling space is well known and isn't going anywhere. The next step, if starting there is to supplement. XYZ and IVM co-exist in a healthy manner. Imagine filling every other box with a growing sphere, a 3D checkerboard. Each ball, fully expanded, touches 12 neighbors as 12 mid-edges.

"All fine and good" you're thinking, "but the hypnotic brainwashing animations described here sounds a lot more like rave party projections, too hallucinogenic for a math-minded audience."  

I'd say fair enough to speculate in that direction, but you really don't have to carry it that far. Surely you too know what it means to daydream in a classroom. 

That's a form of right brain engagement. 

To encourage this faculty is to constructively entrain the imagination, as we do when teaching the value reading in fiction (which is not to "push drugs" (as if any talk of "the imagination" were dangerous witchcraft)) i.e. "you get to watch movies in your head" as a grownup might put it, promising payoff (I understand why watching adults just staring at print, with no pictures, is a turn off until experienced in the first person).

The ratios of which I speak have to do with the so-called voronoi cell encasings around each IVM ball. Every IVM center has its "domain" is another way to but it. These are not Platonics in the strict sense of meeting the "all corners identical" criterion; the faces are diamonds and meet in threes and fours around two sets of corners: those at the corners of a cube (shallow, 3 facets) and those at the corners of an octahedron (sharp, 4 facets). Volume ratios: RD (rhombic dodeca) : Octahedron : Cube :: 6 : 4 : 3.

Why don't those ratios seem as familiar as rain already? Maybe to some of us they do, but the answer is the cube of volume 3, not its canonical most regal volume in the orthodox hierarchy. What dares take its place? A tetrahedron? Heresy!

That's the crack in the pavement a lot of storytellers keep tripping over, in wanting to make believe it's not there. Those of us in the curriculum design business can't deny that tetravolumes are tempting, in some contexts, especially knowing we're not admitting anything. We never have to say "XYZ was wrong" or anything like that. So what's the issue? Descartes is still a hero. We also study his Deficit (720 degrees).

I'm anticipating comments (perhaps by email) that I'm wrestling with ghosts, as none of the MineCraft literature ever talks about "another paradigm" i.e. a space-filling pattern of tetrahedrons and octahedrons. What would that world look like? Now we're talking (and maybe even imagining).